遍历论及其在概率论中的应用

In this project, we try to develop the ergodic theory along with its applications in probability theory. As is well known, many ergodic theorems in ergodic theory have their correspondences in probability theory. For instances,the well-known Poincare's recurrence theorem has a close connection with the recurrence theory of Markov processes (especially, Markov chains); and the classical Birkhoff's indivisual ergodic theorem is in fact a generalization of the strong law of large numbers in probability theory; also, Chacon-Ornstein ratio ergodic theorem has its prototype in Markov chains' theory which is Doeblin's ratio limit theorem...We will start our research at the rang-renewal problem of Markov chains.By try to exploy or redevelop the classical ergodic theorems, we would try to solve some important problems in that field. And surely such a question motivated consideration in probability theory would help us in a good understanding and development of the ergodic theory. In fact we have already developed a simple approach for building strong law of large numbers following the above line. More important results are likely to appear along our research.

本项目研究遍历理论及其在概率论中的应用。遍历理论中的很多遍历定理在概率论中都有相应的经典理论与之对应。例如Poincare回复定理与马氏过程（马氏链）的常返理论密切相关；而经典的Birkhoff逐点遍历定理则是概率论中强大数律的推广；比值遍历定理也可视作以马氏链模型中的Doeblin比值极限定理为原型；凡此等等。..本项目从马氏链的值域更新问题入手，试图利用或改造原有的遍历理论应用于概率论的马氏过程理论中，解决其中的一些重要问题；同时概率论的典型理论和思想方法毫无疑问也将促进我们对遍历理论的理解与发展。沿此思路我们已经发展出了一套建立强大数律的简单方法，相信未来会取得更大进展。

项目期间，以前人在随机游动模型的值域个数的相关问题上的研究结果（强大数律、中心极限定理、重对数律等）与方法为原型，加以遍历论方法重新探索，研究了i.i.d.离散样本的取值个数问题、离散群上暂留的简单随机游动的值域问题、连分数展开式的连分数位取值个数问题等一系列问题，得到了对应的强大数律；由此发现了在正常返的模型中如果平稳分布具有一定规则性，对应模型的值域更新就具有特定的结构以及Power Law。